全波形反演的参数梯度更新公式推导

基于时域

速度 - 密度参数

采用微扰法推导的全过程如下：

时域声波方程为：

$$ -\frac{1}{\rho v^2} \frac{\partial^2 u(v, \rho)}{\partial t^2}+\operatorname{div} \cdot \frac{1}{\rho} \nabla u(v, \rho)=-f \cdot \delta\left(\boldsymbol{x}-\boldsymbol{x}_s\right) $$

当介质参数发生微扰时，波场也会产生微扰，因此扰动后的时域声波方程为

$$ \begin{aligned} &- \frac{1}{(\rho+\delta \rho)(v+\delta v)^2} \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2} \ &+\operatorname{div} \cdot \frac{1}{\rho+\delta \rho} \nabla u(v+\delta v, \rho+\delta \rho)=-f \cdot \delta\left(\boldsymbol{x}-\boldsymbol{x}_s\right) \ & \end{aligned} $$

以上两式相减，并利用下面的近似

$$ \frac{1}{(v+\delta v)^2}=\frac{1}{v^2}-\frac{2 \delta v}{v^3}+o\left(\delta v^2\right), \quad \frac{1}{\rho+\delta \rho}=\frac{1}{\rho}-\frac{\delta \rho}{\rho^2}+o\left(\delta \rho^2\right) $$ 化简后可得下式，其中 $\delta u=u (v+\delta v, \rho+\delta \rho)-u (v, \rho)$

$$ \begin{aligned} & \left(-\frac{1}{\rho v^2} \frac{\partial^2 \delta u}{\partial t^2}+\operatorname{div} \cdot \frac{1}{\rho} \nabla \delta u\right)+\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2} \ & -\operatorname{div} \cdot \frac{\delta \rho}{\rho^2} \nabla u(v+\delta v, \rho+\delta \rho)+o\left(\delta v^2+\delta \rho^2\right)=0 \end{aligned} $$

为了得到波场扰动量和介质参数扰动量之间的关系，上式可以进一步化简为 $$ \begin{aligned} \left(-\frac{1}{\rho v^2} \frac{\partial^2 \delta u}{\partial t^2}+\operatorname{div} \cdot \frac{1}{\rho} \nabla \delta u\right)=&-\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2} \ &+\operatorname{div} \cdot \frac{\delta \rho}{\rho^2} \nabla u(v+\delta v, \rho+\delta \rho)+o\left(\delta v^2+\delta \rho^2\right) \end{aligned} $$ 因此由格林函数定义可知 $$ \begin{aligned} \delta u =& G\ast F \ =& -\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2} \ast G_{0} +\operatorname{div} \cdot \frac{\delta \rho}{\rho^2} \nabla u(v+\delta v, \rho+\delta \rho) \ast G_{0} \end{aligned} $$ 因为误差函数 $$ \begin{aligned} \delta \mathcal{F}(v, \rho)= & \mathcal{F}(v+\delta v, \rho+\delta \rho)-\mathcal{F}(v, \rho) \ =&\sum_{\text {shot }} \int_T \int_{\Omega} \delta \frac{1}{2}\left[u-u_{\mathrm{obs}}\right]^2 \delta\left(\boldsymbol{x}-\boldsymbol{x}r\right) d x d z d t \ = & \sum{\text {shot }} \int_T \int_{\Omega}\left(u-u_{\mathrm{obs}}\right) \delta\left(\boldsymbol{x}-\boldsymbol{x}r\right) \delta u d x d z d t \end{aligned} $$ 所以误差函数关于速度和密度的梯度为 $$ \frac{\delta \mathcal{F}}{\delta m{i}}= \sum_{\text {shot }} \int_T \int_{\Omega}\frac{\delta u}{\delta m_{i}} \delta\left(\boldsymbol{x}-\boldsymbol{x}r\right) \delta u d x d z d t $$ 令 $\delta v, \delta \rho \rightarrow 0$， 即得 $$ \begin{aligned} \frac{\partial \mathcal{F}}{\partial \rho}= & -\sum{\boldsymbol{x}s} \int_T \int{\Omega} \frac{1}{\rho^2} \nabla \phi \cdot \nabla u d x d z d t \ & -\sum_{\boldsymbol{x}s} \int_T \int{\Omega} \frac{1}{\rho^2 v^2} \phi \frac{\partial^2 u}{\partial t^2} \delta \rho d x d z d t \ \frac{\partial \mathcal{F}}{\partial v}= & -\sum_{\boldsymbol{x}s} \int_T \int{\Omega} \frac{2}{\rho v^3} \phi \frac{\partial^2 u}{\partial t^2} d x d z d t \end{aligned} $$

$$ \begin{aligned} \delta \mathcal{F}(v, \rho)= & \sum_{\boldsymbol{x}s} \int_T \int{\Omega} \phi\left(-\frac{1}{\rho v^2} \frac{\partial^2 \delta u}{\partial t^2}+\mathrm{div} \cdot \frac{1}{\rho} \nabla \delta u\right) d x d z d t \ = & \sum_{\boldsymbol{x}s} \int_T \int{\Omega} \phi\left[-\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2}\right] d x d z d t \ & +\sum_{\boldsymbol{x}s} \int_T \int{\Omega} \phi\left[\operatorname{div} \cdot \frac{\delta \rho}{\rho^2} \nabla u(v+\delta v, \rho+\delta \rho)\right] d x d z d t+o\left(\delta v^2+\delta \rho^2\right) . \end{aligned} $$

再对 (8.4.29) 等式右端第二项空间上应用格林公式及边界条件, 可得 $$ \begin{aligned} \delta \mathcal{F}(v, \rho)= & \sum_{\boldsymbol{x}s} \int_T \int{\Omega} \phi\left[-\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2}\right] d x d z d t \ & +\sum_{\boldsymbol{x}s} \int_T d t\left{\left.\frac{\delta \rho}{\rho^2} \frac{\partial u(v+\delta v, \rho+\delta \rho)}{\partial \boldsymbol{n}} \phi\right|{\partial \Omega}\right. \ & \left.-\int_{\Omega} \frac{\delta \rho}{\rho^2} \nabla \phi \cdot \nabla u(v+\delta v, \rho+\delta \rho) d x d z\right}+o\left(\delta v^2+\delta \rho^2\right), \end{aligned} $$

也即 $$ \begin{aligned} \delta \mathcal{F}(v, \rho)= & -\sum_{\boldsymbol{x}s} \int_T \int{\Omega}\left(\frac{\delta \rho}{\rho^2 v^2}+\frac{2 \delta v}{\rho v^3}\right) \phi \frac{\partial^2 u(v+\delta v, \rho+\delta \rho)}{\partial t^2} d x d z d t \ & -\sum_{\boldsymbol{x}s} \int_T \int{\Omega} \frac{\delta \rho}{\rho^2} \nabla \phi \cdot \nabla u(v+\delta v, \rho+\delta \rho) d x d z d t+o\left(\delta v^2+\delta \rho^2\right), \end{aligned} $$